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Consistent manifold representation for topological data analysis
Department of Mathematical Sciences, Fairfax, VA 22030, USA |
For data sampled from an arbitrary density on a manifold embedded in Euclidean space, the Continuous k-Nearest Neighbors (CkNN) graph construction is introduced. It is shown that CkNN is geometrically consistent in the sense that under certain conditions, the unnormalized graph Laplacian converges to the Laplace-Beltrami operator, spectrally as well as pointwise. It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold in the limit of large data. Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale. As applications we derive a new fast clustering algorithm and a method to identify patterns in natural images topologically. Finally, we conjecture that CkNN is topologically consistent, meaning that the homology of the Vietoris-Rips complex (implied by the graph Laplacian) converges to the homology of the underlying manifold (implied by the Laplace-de Rham operators) in the limit of large data.
References:
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M. Belkin and P. Niyogi,
Laplacian eigenmaps for dimensionality reduction and data representation, Neural Computation, 15 (2003), 1373-1396.
doi: 10.1162/089976603321780317. |
[2] |
M. Belkin and P. Niyogi, Convergence of Laplacian eigenmaps, Advances in Neural Information Processing Systems, (2007), 139-136. Google Scholar |
[3] |
T. Berry, J. R. Cressman, Z. G. Ferenček and T. Sauer,
Time-scale separation from diffusion-mapped delay coordinates, SIAM J. Appl. Dyn. Sys, 12 (2013), 618-649.
doi: 10.1137/12088183X. |
[4] |
T. Berry and J. Harlim,
Variable bandwidth diffusion kernels, Appl. Comp. Harmonic Anal., 40 (2016), 68-96.
doi: 10.1016/j.acha.2015.01.001. |
[5] |
T. Berry and T. Sauer,
Local kernels and the geometric structure of data, Appl. Comp. Harmonic Anal., 40 (2016), 439-469.
doi: 10.1016/j.acha.2015.03.002. |
[6] |
T. Berry and D. Giannakis, Spectral exterior calculus, arXiv preprint, arXiv: 1802.01209. Google Scholar |
[7] |
O. Bobrowski, S. Mukherjee, J. E. Taylor et al., Topological consistency via kernel estimation, Bernoulli, 23 (2017), 288–328.
doi: 10.3150/15-BEJ744. |
[8] |
G. Carlsson,
Topology and data, Bulletin of the American Mathematical Society, 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[9] |
J. Chacón,
A population background for nonparametric density-based clustering, Statistical Science, 30 (2015), 518-532.
doi: 10.1214/15-STS526. |
[10] |
K. Chaudhuri, S. Dasgupta, S. Kpotufe and U. von Luxburg,
Consistent procedures for cluster tree estimation and pruning, Information Theory, IEEE Transactions on, 60 (2014), 7900-7912.
doi: 10.1109/TIT.2014.2361055. |
[11] |
A. Cianchi and V. Maz'ya, On the discreteness of the spectrum of the laplacian on noncompact riemannian manifolds, J. Differential Geom., 87 (2011), 469–492, URL http://projecteuclid.org/euclid.jdg/1312998232.
doi: 10.4310/jdg/1312998232. |
[12] |
R. Coifman and S. Lafon,
Diffusion maps, Appl. Comp. Harmonic Anal., 21 (2006), 5-30.
doi: 10.1016/j.acha.2006.04.006. |
[13] |
R. Coifman, S. Lafon, B. Nadler and I. Kevrekidis,
Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comp. Harmonic Anal., 21 (2006), 113-127.
doi: 10.1016/j.acha.2005.07.004. |
[14] |
R. Coifman, Y. Shkolnisky, F. Sigworth and A. Singer,
Graph Laplacian tomography from unknown random projections, IEEE Trans. on Image Proc., 17 (2008), 1891-1899.
doi: 10.1109/TIP.2008.2002305. |
[15] |
M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in Discrete Differential Geometry, Springer, 38 (2008), 287–324.
doi: 10.1007/978-3-7643-8621-4_16. |
[16] |
H. Edelsbrunner and J. Harer, Computational Toplogy: An Introduction, American Mathematical Soc., 2010. |
[17] |
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[18] |
R. Ghrist,
Barcodes: The persistent topology of data, Bulletin of the American Mathematical Society, 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
D. Giannakis and A. J. Majda,
Nonlinear laplacian spectral analysis for time series with intermittency and low-frequency variability, Proceedings of the National Academy of Sciences, 109 (2012), 2222-2227.
doi: 10.1073/pnas.1118984109. |
[20] |
M. Hein, Geometrical aspects of statistical learning theory, Thesis, URL http://elib.tu-darmstadt.de/diss/000673. Google Scholar |
[21] |
M. Hein, Uniform convergence of adaptive graph-based regularization, in Learning Theory, Springer, 4005 (2006), 50–64.
doi: 10.1007/11776420_7. |
[22] |
M. Hein, J.-Y. Audibert and U. Von Luxburg, From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians, in Learning Theory, Springer, 3559 (2005), 470–485.
doi: 10.1007/11503415_32. |
[23] |
A. N. Hirani, Discrete Exterior Calculus, PhD thesis, California Institute of Technology, 2003. |
[24] |
Kallerna, Scale common roach.jpg, 2009, URL https://commons.wikimedia.org/wiki/File:Scale_Common_Roach.JPG, https://commons.wikimedia.org/wiki/File:Scale_Common_Roach.JPG; accessed June 3, 2016; Creative Commons License. Google Scholar |
[25] |
J. Latschev,
Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold, Archiv der Mathematik, 77 (2001), 522-528.
doi: 10.1007/PL00000526. |
[26] |
D. Loftsgaarden, C. Quesenberry et al., A nonparametric estimate of a multivariate density function, The Annals of Mathematical Statistics, 36 (1965), 1049-1051.
doi: 10.1214/aoms/1177700079. |
[27] |
M. Maier, M. Hein and U. Von Luxburg, Cluster identification in nearest-neighbor graphs, in Algorithmic Learning Theory, Springer, 2007, 196–210. Google Scholar |
[28] |
M. Maier, M. Hein and U. von Luxburg, Optimal construction of k-nearest-neighbor graphs for identifying noisy clusters, Theoretical Computer Science, 410 (2009), 1749-1764. Google Scholar |
[29] |
M. Maier, U. Von Luxburg and M. Hein,
How the result of graph clustering methods depends on the construction of the graph, ESAIM: Probability and Statistics, 17 (2013), 370-418.
doi: 10.1051/ps/2012001. |
[30] |
B. Nadler and M. Galun, Fundamental limitations of spectral clustering methods, in Advances in Neural Information Processing Systems 19 (eds. B. Schölkopf, J. Platt and T. Hoffman), MIT Press, Cambridge, MA, 2007. Google Scholar |
[31] |
B. Nadler, S. Lafon, R. Coifman and I. Kevrekidis, Diffusion maps-a probabilistic interpretation for spectral embedding and clustering algorithms, in Principal Manifolds for Data Visualization and Dimension Reduction, Springer, NY, 58 (2008), 238–260.
doi: 10.1007/978-3-540-73750-6_10. |
[32] |
B. Nadler, S. Lafon, I. Kevrekidis and R. Coifman, Diffusion maps, spectral clustering and eigenfunctions of Fokker-Planck operators, in Advances in Neural Information Processing Systems, MIT Press, 2005, 955–962. Google Scholar |
[33] |
A. Ng, M. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, Neur. Inf. Proc. Soc. Google Scholar |
[34] |
P. Niyogi, S. Smale and S. Weinberger,
Finding the homology of submanifolds with high confidence from random samples, Discrete & Computational Geometry, 39 (2008), 419-441.
doi: 10.1007/s00454-008-9053-2. |
[35] |
S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511623783. |
[36] |
D. S. Rufat, Spectral Exterior Calculus and Its Implementation, PhD thesis, California Institute of Technology, 2017, URL http://resolver.caltech.edu/CaltechTHESIS:05302017-094600781. Google Scholar |
[37] |
B. Schölkopf, A. Smola and K. Müller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, 10 (1998), 1299-1319. Google Scholar |
[38] |
E. Schulz and G. Tsogtgerel, Convergence of discrete exterior calculus approximations for Poisson problems, 2016. Google Scholar |
[39] |
Z. Shi, Convergence of laplacian spectra from random samples., arXiv preprint, arXiv: 1507.00151. Google Scholar |
[40] |
A. Singer,
From graph to manifold Laplacian: The convergence rate, Applied and Computational Harmonic Analysis, 21 (2006), 128-134.
doi: 10.1016/j.acha.2006.03.004. |
[41] |
I. Steinwart,
Fully adaptive density-based clustering, The Annals of Statistics, 43 (2015), 2132-2167.
doi: 10.1214/15-AOS1331. |
[42] |
A. Tausz, M. Vejdemo-Johansson and H. Adams, JavaPlex: A research software package for persistent (co)homology, in Proceedings of ICMS 2014 (eds. H. Hong and C. Yap), Lecture Notes in Computer Science 8592, 2014, 129–136, Software available at http://appliedtopology.github.io/javaplex/. Google Scholar |
[43] |
D. G. Terrell and D. W. Scott,
Variable kernel density estimation, Annals of Statistics, 20 (1992), 1236-1265.
doi: 10.1214/aos/1176348768. |
[44] |
D. Ting, L. Huang and M. Jordan, An analysis of the convergence of graph Laplacians, in Proceedings of the 27th International Conference on Machine Learning (ICML), 2010. Google Scholar |
[45] |
N. G. Trillos, M. Gerlach, M. Hein and D. Slepcev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace-Beltrami operator, arXiv preprint, arXiv: 1801.10108. Google Scholar |
[46] |
N. G. Trillos and D. Slepčev, A variational approach to the consistency of spectral clustering, Applied and Computational Harmonic Analysis, 45 (2018), 239-281. Google Scholar |
[47] |
U. Von Luxburg,
A tutorial on spectral clustering, Statistics and Computing, 17 (2007), 395-416.
doi: 10.1007/s11222-007-9033-z. |
[48] |
U. Von Luxburg, M. Belkin and O. Bousquet,
Consistency of spectral clustering, Annals of Statistics, 36 (2008), 555-586.
doi: 10.1214/009053607000000640. |
[49] |
L. Zelnick-Manor and P. Perona, Self-tuning spectral clustering, Adv. Neur. Inf. Proc. Sys. (2005) Google Scholar |
show all references
References:
[1] |
M. Belkin and P. Niyogi,
Laplacian eigenmaps for dimensionality reduction and data representation, Neural Computation, 15 (2003), 1373-1396.
doi: 10.1162/089976603321780317. |
[2] |
M. Belkin and P. Niyogi, Convergence of Laplacian eigenmaps, Advances in Neural Information Processing Systems, (2007), 139-136. Google Scholar |
[3] |
T. Berry, J. R. Cressman, Z. G. Ferenček and T. Sauer,
Time-scale separation from diffusion-mapped delay coordinates, SIAM J. Appl. Dyn. Sys, 12 (2013), 618-649.
doi: 10.1137/12088183X. |
[4] |
T. Berry and J. Harlim,
Variable bandwidth diffusion kernels, Appl. Comp. Harmonic Anal., 40 (2016), 68-96.
doi: 10.1016/j.acha.2015.01.001. |
[5] |
T. Berry and T. Sauer,
Local kernels and the geometric structure of data, Appl. Comp. Harmonic Anal., 40 (2016), 439-469.
doi: 10.1016/j.acha.2015.03.002. |
[6] |
T. Berry and D. Giannakis, Spectral exterior calculus, arXiv preprint, arXiv: 1802.01209. Google Scholar |
[7] |
O. Bobrowski, S. Mukherjee, J. E. Taylor et al., Topological consistency via kernel estimation, Bernoulli, 23 (2017), 288–328.
doi: 10.3150/15-BEJ744. |
[8] |
G. Carlsson,
Topology and data, Bulletin of the American Mathematical Society, 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[9] |
J. Chacón,
A population background for nonparametric density-based clustering, Statistical Science, 30 (2015), 518-532.
doi: 10.1214/15-STS526. |
[10] |
K. Chaudhuri, S. Dasgupta, S. Kpotufe and U. von Luxburg,
Consistent procedures for cluster tree estimation and pruning, Information Theory, IEEE Transactions on, 60 (2014), 7900-7912.
doi: 10.1109/TIT.2014.2361055. |
[11] |
A. Cianchi and V. Maz'ya, On the discreteness of the spectrum of the laplacian on noncompact riemannian manifolds, J. Differential Geom., 87 (2011), 469–492, URL http://projecteuclid.org/euclid.jdg/1312998232.
doi: 10.4310/jdg/1312998232. |
[12] |
R. Coifman and S. Lafon,
Diffusion maps, Appl. Comp. Harmonic Anal., 21 (2006), 5-30.
doi: 10.1016/j.acha.2006.04.006. |
[13] |
R. Coifman, S. Lafon, B. Nadler and I. Kevrekidis,
Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comp. Harmonic Anal., 21 (2006), 113-127.
doi: 10.1016/j.acha.2005.07.004. |
[14] |
R. Coifman, Y. Shkolnisky, F. Sigworth and A. Singer,
Graph Laplacian tomography from unknown random projections, IEEE Trans. on Image Proc., 17 (2008), 1891-1899.
doi: 10.1109/TIP.2008.2002305. |
[15] |
M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in Discrete Differential Geometry, Springer, 38 (2008), 287–324.
doi: 10.1007/978-3-7643-8621-4_16. |
[16] |
H. Edelsbrunner and J. Harer, Computational Toplogy: An Introduction, American Mathematical Soc., 2010. |
[17] |
S. Fazel, Zebra in mikumi.jpg, 2012, URL https://commons.wikimedia.org/wiki/File:Zebra_in_Mikumi.JPG, https://commons.wikimedia.org/wiki/File:Zebra_in_Mikumi.JPG; accessed June 3, 2016; Creative Commons License. Google Scholar |
[18] |
R. Ghrist,
Barcodes: The persistent topology of data, Bulletin of the American Mathematical Society, 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[19] |
D. Giannakis and A. J. Majda,
Nonlinear laplacian spectral analysis for time series with intermittency and low-frequency variability, Proceedings of the National Academy of Sciences, 109 (2012), 2222-2227.
doi: 10.1073/pnas.1118984109. |
[20] |
M. Hein, Geometrical aspects of statistical learning theory, Thesis, URL http://elib.tu-darmstadt.de/diss/000673. Google Scholar |
[21] |
M. Hein, Uniform convergence of adaptive graph-based regularization, in Learning Theory, Springer, 4005 (2006), 50–64.
doi: 10.1007/11776420_7. |
[22] |
M. Hein, J.-Y. Audibert and U. Von Luxburg, From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians, in Learning Theory, Springer, 3559 (2005), 470–485.
doi: 10.1007/11503415_32. |
[23] |
A. N. Hirani, Discrete Exterior Calculus, PhD thesis, California Institute of Technology, 2003. |
[24] |
Kallerna, Scale common roach.jpg, 2009, URL https://commons.wikimedia.org/wiki/File:Scale_Common_Roach.JPG, https://commons.wikimedia.org/wiki/File:Scale_Common_Roach.JPG; accessed June 3, 2016; Creative Commons License. Google Scholar |
[25] |
J. Latschev,
Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold, Archiv der Mathematik, 77 (2001), 522-528.
doi: 10.1007/PL00000526. |
[26] |
D. Loftsgaarden, C. Quesenberry et al., A nonparametric estimate of a multivariate density function, The Annals of Mathematical Statistics, 36 (1965), 1049-1051.
doi: 10.1214/aoms/1177700079. |
[27] |
M. Maier, M. Hein and U. Von Luxburg, Cluster identification in nearest-neighbor graphs, in Algorithmic Learning Theory, Springer, 2007, 196–210. Google Scholar |
[28] |
M. Maier, M. Hein and U. von Luxburg, Optimal construction of k-nearest-neighbor graphs for identifying noisy clusters, Theoretical Computer Science, 410 (2009), 1749-1764. Google Scholar |
[29] |
M. Maier, U. Von Luxburg and M. Hein,
How the result of graph clustering methods depends on the construction of the graph, ESAIM: Probability and Statistics, 17 (2013), 370-418.
doi: 10.1051/ps/2012001. |
[30] |
B. Nadler and M. Galun, Fundamental limitations of spectral clustering methods, in Advances in Neural Information Processing Systems 19 (eds. B. Schölkopf, J. Platt and T. Hoffman), MIT Press, Cambridge, MA, 2007. Google Scholar |
[31] |
B. Nadler, S. Lafon, R. Coifman and I. Kevrekidis, Diffusion maps-a probabilistic interpretation for spectral embedding and clustering algorithms, in Principal Manifolds for Data Visualization and Dimension Reduction, Springer, NY, 58 (2008), 238–260.
doi: 10.1007/978-3-540-73750-6_10. |
[32] |
B. Nadler, S. Lafon, I. Kevrekidis and R. Coifman, Diffusion maps, spectral clustering and eigenfunctions of Fokker-Planck operators, in Advances in Neural Information Processing Systems, MIT Press, 2005, 955–962. Google Scholar |
[33] |
A. Ng, M. Jordan and Y. Weiss, On spectral clustering: Analysis and an algorithm, Neur. Inf. Proc. Soc. Google Scholar |
[34] |
P. Niyogi, S. Smale and S. Weinberger,
Finding the homology of submanifolds with high confidence from random samples, Discrete & Computational Geometry, 39 (2008), 419-441.
doi: 10.1007/s00454-008-9053-2. |
[35] |
S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511623783. |
[36] |
D. S. Rufat, Spectral Exterior Calculus and Its Implementation, PhD thesis, California Institute of Technology, 2017, URL http://resolver.caltech.edu/CaltechTHESIS:05302017-094600781. Google Scholar |
[37] |
B. Schölkopf, A. Smola and K. Müller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, 10 (1998), 1299-1319. Google Scholar |
[38] |
E. Schulz and G. Tsogtgerel, Convergence of discrete exterior calculus approximations for Poisson problems, 2016. Google Scholar |
[39] |
Z. Shi, Convergence of laplacian spectra from random samples., arXiv preprint, arXiv: 1507.00151. Google Scholar |
[40] |
A. Singer,
From graph to manifold Laplacian: The convergence rate, Applied and Computational Harmonic Analysis, 21 (2006), 128-134.
doi: 10.1016/j.acha.2006.03.004. |
[41] |
I. Steinwart,
Fully adaptive density-based clustering, The Annals of Statistics, 43 (2015), 2132-2167.
doi: 10.1214/15-AOS1331. |
[42] |
A. Tausz, M. Vejdemo-Johansson and H. Adams, JavaPlex: A research software package for persistent (co)homology, in Proceedings of ICMS 2014 (eds. H. Hong and C. Yap), Lecture Notes in Computer Science 8592, 2014, 129–136, Software available at http://appliedtopology.github.io/javaplex/. Google Scholar |
[43] |
D. G. Terrell and D. W. Scott,
Variable kernel density estimation, Annals of Statistics, 20 (1992), 1236-1265.
doi: 10.1214/aos/1176348768. |
[44] |
D. Ting, L. Huang and M. Jordan, An analysis of the convergence of graph Laplacians, in Proceedings of the 27th International Conference on Machine Learning (ICML), 2010. Google Scholar |
[45] |
N. G. Trillos, M. Gerlach, M. Hein and D. Slepcev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace-Beltrami operator, arXiv preprint, arXiv: 1801.10108. Google Scholar |
[46] |
N. G. Trillos and D. Slepčev, A variational approach to the consistency of spectral clustering, Applied and Computational Harmonic Analysis, 45 (2018), 239-281. Google Scholar |
[47] |
U. Von Luxburg,
A tutorial on spectral clustering, Statistics and Computing, 17 (2007), 395-416.
doi: 10.1007/s11222-007-9033-z. |
[48] |
U. Von Luxburg, M. Belkin and O. Bousquet,
Consistency of spectral clustering, Annals of Statistics, 36 (2008), 555-586.
doi: 10.1214/009053607000000640. |
[49] |
L. Zelnick-Manor and P. Perona, Self-tuning spectral clustering, Adv. Neur. Inf. Proc. Sys. (2005) Google Scholar |













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